WILLIAMS. — GEOMETRY ON RULED QUARTIC SURFACES. 31 



are points of a^ ; if, then, any generator meets S'^") in an additional point 

 it must lie on *SM, and therefore if any generator has on it a point of the 

 residual it lies on /S^'') and is itself a part of the residual ; therefore, when 



a = - the residual consists entirely of generators which are a in number, 



since the residual is of order a ; and if the curve has no multiple points 



on T, there are - pairs of generators that pass through the - points 



where «„ meets T. 



We have shown then that every twisted curve of order a can be cut 

 out by an S'^"^ such that the residual will consist of curves of orders less 

 than a, and it therefore follows, as on page 29, that formula (1) holds 

 for every curve on the scroll. 



IV. QuARTic Scroll, with two Double Linear Directors and 

 WITH A Double Generator, S (I2, I2) 2). (Caylet's Second 

 Species, ^'(Ij, lo, 4).) 



1. Let us call the double linear directors D and U ; they do not 

 intersect, and a plane through either of them cuts out also two generators 

 that intersect in the point where the plane meets the other director, 

 i. e. any generator A meets a definite generator B on D and another 

 definite generator E on Z)', so that A and B lie in a plane through ZX 

 and A and E lie in a plane through D, while B and E do not meet. In 

 a special form of this scroll four generators may form a gauch-quadrilat- 

 eral having two vertices on each double director, e. g. taking the gener- 

 ators above, if A and B meet D at the point P, A and E meet D' at the 

 point R, B meets D' at the point S, and E meets D at the point Q, the 

 scroll may be of such a form that a generator i^will pass through Q and 

 S, as can easily be shown analytically. It is also very probable that 

 there are special forms of this scroll on which any even number of gen- 

 erators, greater than four, form a gauch-polygon, but it is not the purpose 

 of this paper to discuss these special forms. The double generator, which 

 we will denote by G, arises from the fact that the plane quartic direct- 

 ing curve has three double points, one on each of the double directors 

 and one through which G passes ; a plane through G and either double 

 director does not meet the scroll again. 



2. Proof of Theorem I. — On any quartic scroll where the double 

 curve is a twisted cubic, either proper or degenerate, we can prove 

 Theorem I by passing a quadric through this twisted cubic. In the 



